More generally, every simplicial topological space whose topology is degreewise detectable by Euclidean topologies canonically identifies with a Euclidean-topological ∞\infty-groupoid. Various constructions with simplicial toppological spaces find their natural home in this (∞,1)-topos. For instance

But by the above proposition we have that before hypercompletionSh(∞,1)(CartSptop)Sh_{(\infty,1)}(CartSp_{top}) is cohesive. This means that it is in particular a local (∞,1)-topos. By the discussion there, this means that it already coincides with its hypercompletion, Sh(∞,1)(CartSptop)≃Sh^(∞,1)(CartSptop)Sh_{(\infty,1)}(CartSp_{top}) \simeq \hat Sh_{(\infty,1)}(CartSp_{top}).

Proof

While the model structures on simplicial presheaves over both sites present the same (∞,1)-category, they lend themselves to different computations:

the model structure over CartSptopCartSp_{top} has more fibrant objects and hence fewer cofibrant objects, while the model structure over MfdMfd has more cofibrant objects and fewer fibrant objects. More specifically:

Proposition

Let X∈[Mfdop,sSet]X \in [Mfd^{op}, sSet] be an object that is globally fibrant , separated and locally trivial, meaning that

is a full and faithful functor (over each object U∈MfdU \in Mfd): it includes the single object of W¯G\bar W G as the trivial GG-principal bundle. The automorphism of the single object in W¯G\bar W G over UU are GG-valued continuous functions on UU, which are precisely the automorphisms of the trivial GG-bundle. Therefore this inclusion is full and faithful, the presheaf W¯G\bar W G is a separated prestack.

Moreover, it is locally trivial: every Cech cocycle for a GG-bundle over a Cartesian space is equivalent to the trivial one. Equivalently, also π0GBund(ℝn)≃*\pi_0 G Bund(\mathbb{R}^n) \simeq *.

Therefore W¯G\bar W G, when restricted CartSptopCartSp_{top}, does become a fibrant object in [CartSptopop,sSet]proj,loc[CartSp_{top}^{op}, sSet]_{proj,loc}.

On the other hand, let X∈MfdX \in Mfd be any non-contractible manifold. Since in the projective model structure on simplicial presheaves every representable is cofibrant, this is a cofibrant object in [Mfdop,sSet]proj,loc[Mfd^{op}, sSet]_{proj,loc}. However, it fails to be cofibrant in [CartSptopop,sSet]proj,loc[CartSp_{top}^{op}, sSet]_{proj,loc}. Instead, there a cofibrant replacement is given by the Cech nerveC({Ui})C(\{U_i\}) of any good open cover{Ui→X}\{U_i \to X\}.

In the first case we need to construct the fibrant replacement GBundG Bund. This amounts to computing GG-cocycles = GG-bundles over all manifolds and then evaluate on the given one, XX, by the 2-Yoneda lemma.

In the second case however we cofibrantly replace XX by a good open cover, and then find the Cech cocycles with coefficients in GG on that.

For ordinary GG-bundles the difference between the two computations may be irrelevant in practice, because ordinary GG-bundles are very well understood. However for more general coefficient objects, for instance general topological simplicial groups GG, the first approach requires to find the full ∞-stackification to the ∞-stack of all principal ∞-bundles, while the second approach requires only to compute specific coycles over one specific base object. In practice the latter is often all that one needs.

Proof

By the discussion at ∞-cohesive site we have an equivalence Π(−)≃𝕃lim→\Pi(-) \simeq \mathbb{L} \lim_\to to the derived functor of the sSet-colimit functor lim→:[CartSpop,sSet]proj,loc→sSetQuillen\lim_\to : [CartSp^{op}, sSet]_{proj,loc} \to sSet_{Quillen}.

Remark

We may regard Top itself as a cohesive (∞,1)-topos. (ΠTop⊣DiscTop⊣ΓTop⊣coDiscTop)Top→≃∞Grpd(\Pi_{Top}\dashv Disc_{Top} \dashv \Gamma_{Top} \dashv coDisc_{Top}) Top \stackrel{\simeq}{\to} \infty Grpd. This is discussed at discrete ∞-groupoid.

Proof

Write QQ for Dugger’s cofibrant replacement functor on [CartSpop,sSet]proj,loc[CartSp^{op}, sSet]_{proj,loc} (discussed at model structure on simplicial presheaves). On a simplicially constant simplicial presheaf XX it is given by

which is the simplicial presheaf that over any ℝn∈CartSp\mathbb{R}^n \in CartSp takes as value the diagonal of the bisimplicial set whose (n,r)(n,r)-entry is ∐U0→⋯→Un→XkCartSptop(ℝn,U0)\coprod_{U_0 \to \cdots \to U_n \to X_k} CartSp_{top}(\mathbb{R}^n,U_0).

Since coends are special colimits, the colimit functor itself commutes with them and we find

is a weak equivalence in [CartSpop,sSet]proj[CartSp^{op}, sSet]_{proj}. This implies the claim with prop. 3.

Remark

Typically one is interested in mapping out of Π(X)\mathbf{\Pi}(X). While it is clear that DiscSingXDisc Sing X is cofibrant in [CartSpop,sSet]proj,loc[CartSp^{op}, sSet]_{proj,loc}, it is harder to determine the necessary resolutions of SingX\mathbf{Sing}X.

Proposition

Let GG be a well-pointed simplicial topological group degreewise in TopMfd. Then the (∞,1)(\infty,1)-functor Π:ETop∞Grpd→∞Grpd\Pi : \mathrm{ETop}\infty\mathrm{Grpd} \to \infty \mathrm{Grpd} preserves homotopy fibers of all morphisms of the form X→BGX \to \mathbf{B}G that are presented in [CartSptopop,sSet]proj[\mathrm{CartSp}_{\mathrm{top}}^{\mathrm{op}}, \mathrm{sSet}]_{proj} by morphism of the form X→W¯GX \to \bar W G with XX fibrant.

Proof

Notice that since (∞,1)-sheafification preserves finite (∞,1)-limits we may indeed discuss the homotopy fiber in the global model structure on simplicial presheaves.

Write QX→≃XQ X \stackrel{\simeq}{\to} X for the global cofibrant resolution given by QX:[n]↦∐{Ui0→⋯→Uin→Xn}Ui0Q X : [n] \mapsto \coprod_{\{U_{i_0} \to \cdots \to U_{i_n} \to X_n\}} U_{i_0}, where the UikU_{i_k} range over CartSptop\mathrm{CartSp}_{\mathrm{top}} . (Discussed at model structure on simplicial presheaves – cofibrant replacement. ) This has degeneracies splitting off as direct summands, and hence is a good simplicial topological space that is degreewise in TopMfd. Consider then the pasting of two pullback diagrams of simplicial presheaves

By the discussion at geometric realization of simplicial topological spaces we have that the rightmost vertical morphism is a fibration in [CartSptopop,sSet]proj[CartSp_{top}^{op}, sSet]_{proj}. Since fibrations are stable under pullback, the middle vertical morphism is also a fibration (as is the leftmost one). Since the global model structure is a right proper model category it follows then that also the top left horizontal morphism is a weak

Since the square on the right is a pullback of fibrant objects with one morphism being a fibration, PP is a presentation of the homotopy fiber of X→W¯GX \to \bar W G. Hence so is P′P', which is moreover the pullback of a diagram of good simplicial spaces.

This is a UU-parameterized family of objects of AA together with a U0U_0-parameterized family of morphisms of AA associated to the pairs of points (s,t)∈U(s,t) \in U, hence to the “straight paths” from ss to tt. At the next stage for every triangle of such straight path a 2-morphism is thrown in, and so on. So SingU\mathbf{Sing}U indeed is an ∞\infty-groupoid of paths in UU.

Since every representable UU is cofibrant in [Cop,sSet]proj[C^{op}, sSet]_{proj} and since U→SingUU \to \mathbf{Sing}U is a cofibration by the small object argument, we have that SingU\mathbf{Sing}U is cofibrant in [Cop,sSet]proj[C^{op}, sSet]_{proj} for all UU. This means that also Sing(−)\mathbf{Sing}(-) is cofibrant in [C,[Cop,sSet]pro]inj[C, [C^{op}, sSet]_{pro}]_{inj}. Since ∫C(−)⋅(−)\int^C (-) \cdot (-) is a left Quillen bifunctor it follows that ∫C(−)⋅Sing\int^C (-)\cdot \mathbf{Sing} is a left Quillen functor. Hence it preserves cofibrations and acyclic cofibrations.

This establishes that Sing\mathbf{Sing} is a left simplicial Quillen functor on [Cop,sSet]proj[C^{op}, sSet]_{proj}.

Since this is a left proper model category we have by the discussion at simplicial Quillen adjunction that for showing that this does descend to the local model structure it is sufficient to check that the right adjoint preserves local fibrant objects. Which, in turn, is implied if Sing\mathbf{Sing} send covering Cech nerves to weak equivalences.

Let therefore C(∐iUi→U)C(\coprod_i U_i \to U) be the Cech nerve of a covering family in the siteCC. We may write this as the coend

where by assumption on the ∞-connected siteCC all the Ui0,⋯,inU_{i_0, \cdots, i_n} are representable. By precomposing the projection C(∐iUi)→XC(\coprod_i U_i) \to X with the objectwise Bousfield-Kan map that replaces the simplices with the fat simplexΔ:Δ→sSet\mathbf{\Delta} : \Delta \to sSet, we get the morphisms

Here the first map is an objectwise weak equivalence by Bousfield-Kan (see the examples at Reedy model structure for details). Hence by 2-out-of-3 we may equivalently check that Sing\mathbf{Sing} sends these morphisms to weak equivalences in [Cop,sSet]proj[C^{op}, sSet]_{proj}.

Since Sing\mathbf{Sing} commutes with all colimits and hence coends the result of applying it to this morphism is

the functor ∫ΔΔ⋅(−)\int^\Delta \mathbf{\Delta} \cdot (-) is left Quillen and hence preserves weak equivalences between cofibrant objects (by the factorization lemma), such as the morphisms SingU→≃*\mathbf{Sing}U \stackrel{\simeq}{\to} *. Therefore we have a commuting diagram

with weak equivalences in [Cop,sSet]proj[C^{op}, sSet]_{proj} as indicated: the top morphism is a weak equivalence by the argument just given, the bottom one by the small object argument-construction of Sing\mathbf{Sing} and the right vertical morphism is a weak equivalence by the assumption on an ∞-connected site. It follows by 2-out-of-3 that also the left vertical morphism is a weak equivalence.

Proof

By definition we have that ΠdR\mathbf{\Pi}_{dR} is the (∞,1)-pushoutΠ(X)∐X*\mathbf{\Pi}(X) \coprod_X * in Sh(∞,1)(C)Sh_{(\infty,1)}(C). By the above proposition we have a cofibrant presentation of the pushout diagram as indicated (all three objects cofibrant, at least one of the two morphisms a cofibration). By the general discussion at homotopy colimit the ordinary pushout of that diagram does compute the (∞,1)-colimit.

This provides a useful resolution of topological spaces that often helps to disentangle the two different roles played by a topological space: on the one hand as a model for an ∞-groupoid, in the other as a locale.

Let SPSh(Diff)IlocSPSh(Diff)^{loc}_I be furthermore the left Bousfield localization at the set of projection morphisms out of products of the form X×ℝ→XX \times \mathbb{R} \to X for all X∈DiffX \in Diff. The ∞\infty-stacks that are local objects with respect to these morphisms are the homotopy invariant∞\infty-stacks, so this localization models the (∞,1)-topos of homotopy invariant ∞\infty-stacks on DiffDiff.